this is my first question and I'm not a mathematician so I hope to not make any confusion with terminology. The question: suppose I have a "unitary" (edge length 1) hyper-cube in dimension n. Now I apply as a linear transformation a matrix $A$ of dimension $m \times n$ with $m<n$. This will transform and project the hyper-cube in a space of lower dimension $m$. How can I find the convex hull polytope of the hyper-cube in the new $\mathbb{R}^m$ in terms of LMI or hyper-planes delimiting the convex hull? I know I could simply project all the vertexes using $A$, and then use a convex hull algorithm as https://www.cs.princeton.edu/~chazelle/pubs/ConvexHullAlgorithm.pdf, but I was hoping to find it in closed form due to the particular structure of the hyper-cube and the linear transformation. (Easy) Example: $n=3$, $m=2$, $A=\begin{bmatrix}0 & 1 & -1 \\1 & 0 & -1 \end{bmatrix}$, the cube defined by vertex $(0,0,0),(0,0,1),(0,1,0),(0,1,1),(1,0,0),(1,0,1),(1,1,0),(1,1,1)$.
PS: I noticed that planes in $\mathbb{R}^3$ defining the convex hull of the cube (the facets of the cube) cannot be mapped by $A$ into lines in $\mathbb{R}^2$ and use those as convex hull (why??).